Dynamic and Static FE Model

In this section the governing equations of the time harmonic and static FEM models are introduced. It is important to examine them in order to compre-

CHAPTER 12. FEM MODEL 75 hend how the induced eddy currents are determined. Coupled to these models are the analytical circuit models that model the impedance of the ends in the case of the transient model and the whole conductor in the case of the dynamic model. To begin let's look at the eld magnetisation characterisation due to a PM which is given by

B = ∇ × A = µH + M (12.3.1) where A is the ux vector potential, µ is the magnetic material permeability, H is the applied eld intensity, M is the magnetisation due to the permanent magnets.

12.3.1 Dynamic Governing Equations

In the time harmonic model, the vector potential formulation assumes eddy and source currents expressed by

∇ × 1 µ∇ × A  = σ  −∂ ∂tA − ∇φ  + 1 µ∇ × M (12.3.2) where the term on the right hand side of the equation represents the conduction current density, which is also known as the eddy current expressed as follows:

J = σE = σ  −∂ ∂tA − ∇φ  (12.3.3) where E represents the electric eld density, ∇φ is the electric potential due to the spatial curvature or spatial variation of the electric potential in the magnetic eld, ∂

∂tAis the time variation of the vector potential. Additionally,

when linearity is assumed the current density may be assumed as follows J = σ (−jωA − ∇φ) (12.3.4) where ω is the eld operating frequency, with j indicating that the quantity is sinusoidal.

12.3.2 Static Governing Equations

The static model assumes the following vector potential formulation, 1

µ∇ × ∇ × A = J + 1

µ∇ × M (12.3.5)

where J is the prescribed source current density, and the second term on the right hand side of the equation is the equivalent magnetic current density.

CHAPTER 12. FEM MODEL 76 In the analytical induced eddy current formulation, it appears that the model assumes linear properties and neglects the spatial variation of the ux i.e. assumes the ∇φ term of the eddy current formulation to be zero. The cur- rent density J is prescribed in Eq. 12.3.5 is determined by using the models and methods discussed in Chapter 8 and 9.

Slip (%) 0 5 10 15 20 25 Current (A) 0 500 1000 1500 2000 2500 3000 Static Dynamic (a)

Angular position (deg)

0 90 180 270 360 Current (A) -1000 -500 0 500 1000 Dynamic Static (b)

Figure 12.3: Induced current (a) peak current magnitude and (b) the current waveforms obtained from the spoke-mount PM machine.

Harmonic order 0 5 10 15 20 FFT #105 -1 0 1 2 3 (a) Harmonic order 0 5 10 15 20 FFT #105 -1 0 1 2 3 (b)

Figure 12.4: Induced current harmonics from the (a) static and (b) transient FEM model for the spoke-mount PM machine.

The discrepancies observed in the machine performance prediction obtained from the static and dynamic model can be justied by the time harmonic components of the electrical elds. For example shown in Fig. 12.4a are the induced current time harmonics assumed in the static FEM model and in Fig. 12.4b are the time harmonics that emerge from the time harmonic FEM so- lution. There seems to be prominent 3rd and 5th harmonic components that arise from the time harmonic FEM solution, and this could be the result of

CHAPTER 12. FEM MODEL 77 the discrepancy in the induced current magnitude and induced torque that has been highlighted in earlier sections/chapters.

12.4 Results

The induced currents from the static and dynamic FEM model are shown in Fig. 12.3. When a slip frequency sweep is performed the induced peak current value shown in Fig. 12.3a are obtained. The induced currents from the FEM models are in agreement with some discrepancies that are assumed to be due to the model formulation assumptions. Fig. 12.3b illustrates the harmonic distortion that is observed in the time harmonic induced current plot and how that is dierent to the static induced current, that is assumed purely sinu- soidal. Also observed are the discrepancies in the peak induced current. The highlighted discrepancies are assumed to be responsible for the discrepancies observed in the torque performance predictions as will be demonstrated in Chapter 13.

12.5 Summary

The core FEM governing equations for the static and dynamic model are pre- sented in order to better understand, appreciate and highlight assumption made by each model and the reasons as to why one would observe discrepan- cies in results obtained from the FEM model predictions. It is shown that in the static FEM model, the induced currents are specied as source currents, which are determined using the analytical models of Chapter 8 and the anal- ysis techniques of Chapter 9. In the dynamic FEM the induced currents are determined by the non linear time variation of the ux and spatial variation of the ux, resulting in the induced armature voltage and currents. The har- monics on the assumed induced current in static model and resulting dynamic induced currents from the dynamic model are highlighted with dierences in shape and amplitude of the current waveform.

Chapter 13

Design Optimization

13.1 Introduction

The slip-PMC's outer PM rotor is optimized for the xed inner wound rotor size on the preexisting prototype shown in Fig. 13.9a. The optimization is performed for the spoke-mounted PM rotor conguration in Fig. 8.2. One would like to see how the variation in magnet width and height aects the PM mass, active mass and performance of the machine.

The dimensional parameters that have to be optimized in the design are [x] =  h w  , (13.1.1)

where h is the magnet height and w is the magnet width as indicated in Fig. 13.1.

h

w

Figure 13.1: Optimisation dimensions.